2013/12/05 by M. Briskin, Briskin, M., F. Pakovich +3
Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #math.CA
paper · pdf · doi:10.48550/arxiv.1312.1609
This version is identical to the first one. The replacement is due to the fact that by mistake as a second version another paper was downnloaded. The paper was published by "Qual. Theory Dyn. Syst" in 2014
arxiv created 2018/01/08 · arxiv updated 2018/01/09
The Abel differential equation y'=p(x)y3 + q(x) y2 with meromorphic coefficients p,q is said to have a center on [a,b] if all its solutions, with the initial value y(a) small enough, satisfy the condition y(a)=y(b). The problem of giving conditions on (p,q,a,b) implying a center for the Abel equation is analogous to the classical Poincaré Center-Focus problem for plane vector fields. Following [3,4,8,9] we say that Abel equation has a "parametric center" if for each ε ∈ \mathbb C the equation y'=p(x)y3 + ε q(x) y2 has a center. In the present paper we use recent results of [15,6 to show show that for a polynomial Abel equation parametric center implies strong "composition" restriction on p and q. In particular, we show that for °p,q ≤ 10 parametric center is equivalent to the so-called "Composition Condition" (CC) on p,q. Second, we study trigonometric Abel equation, and provide a series of examples, generalizing a recent remarkable example given in [8], where certain moments of p,q vanish while (CC) is violated.